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Sec 2.6 Geometry – Triangle Proofs Name: COMMON POTENTIAL REASONS FOR PROOFS Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Definition of Midpoint: The point that divides a segment into two congruent segments. Definition of Angle Bisector: The ray that divides an angle into two congruent angles. Definition of Perpendicular Lines: Lines that intersect to form right angles or 90° Definition of Supplementary Angles: Any two angles that have a sum of 180° Definition of a Straight Line: An undefined term in geometry, a line is a straight path that has no thickness and extends forever. It also forms a straight angle which measures 180° Reflexive Property of Equality: any measure is equal to itself (a = a) Reflexive Property of Congruence: any figure is congruent to itself ( Addition Property of Equality: if a = b, then a + c = b + c Subtraction Property of Equality: if a = b, then a – c = b – c Multiplication Property of Equality: if a = b, then ac = bc ≅ ) Division Property of Equality: if a = b, then Transitive Property: if a = b & b =c then a = c OR if a b & b c then a c. Segment Addition Postulate: If point B is between Point A and C then AB + BC = AC Angle Addition Postulate: If point S is in the interior of PQR, then mPQS + mSQR = mPQR Side – Side – Side Postulate (SSS) : If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Side – Angle – Side Postulate (SAS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Angle – Side – Angle Postulate (ASA): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Angle – Angle – Side Postulate (AAS) : If two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the triangles are congruent Hypotenuse – Leg Postulate (HL): If a hypotenuse and a leg of one right triangle are congruent to a hypotenuse and a leg of another right triangle, then the triangles are congruent Right Angle Theorem (R.A.T.): All right angles are congruent. Vertical Angle Theorem (V.A.T.): Vertical angles are congruent. Triangle Sum Theorem: The three angles of a triangle sum to 180° Linear Pair Theorem: If two angles form a linear pair then they are adjacent and are supplementary. Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. Alternate Interior Angle Theorem (and converse): Alternate interior angles are congruent if and only if the transversal that passes through two lines that are parallel. Alternate Exterior Angle Theorem (and converse): Alternate exterior angles are congruent if and only if the transversal that passes through two lines that are parallel. Corresponding Angle Theorem (and converse) : Corresponding angles are congruent if and only if the transversal that passes through two lines that are parallel. Same-Side Interior Angles Theorem (and converse) : Same Side Interior Angles are supplementary if and only if the transversal that passes through two lines that are parallel. Pythagorean Theorem (and converse): A triangle is right triangle if and only if the given the length of the legs a and b and hypotenuse c have the relationship a2+b2 = c2 Isosceles Triangle Theorem (and converse): A triangle is isosceles if and only if its base angles are congruent. Triangle Mid-segment Theorem: A mid-segment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. CPCTC: Corresponding Parts of Congruent Triangles are Congruent by definition of congruence. 1. Tell which of the following triangle provide enough information to show that they must be congruent. If they are congruent, state which theorem suggests they are congruent (SAS, ASA, SSS, AAS, HL) and write a congruence statement. Circle one of the following: SSS SAS ASA AAS HL Not Enough Information Circle one of the following: SAS ASA AAS HL Not Enough Information Circle one of the following: SAS ASA AAS HL Not Enough Information Circle one of the following: SAS ASA AAS Congruence Statement if necessary: ASA AAS HL Circle one of the following: SSS SAS ASA AAS HL SAS HL Not Enough Information ASA AAS HL SAS Not Enough Information Not Enough Information ASA AAS HL Not Enough Information Congruence Statement if necessary: M.Winking SAS ASA AAS HL Not Enough Information HL Not Enough Information HL Not Enough Information Circle one of the following: SSS SAS ASA AAS Circle one of the following: SSS SAS ASA AAS Congruence Statement if necessary: Circle one of the following: SSS SSS Congruence Statement if necessary: Circle one of the following: SSS Circle one of the following: Congruence Statement if necessary: Congruence Statement if necessary: Congruence Statement if necessary: SSS SAS Congruence Statement if necessary: Congruence Statement if necessary: SSS SSS Not Enough Information Congruence Statement if necessary: Congruence Statement if necessary: SSS Circle one of the following: Circle one of the following: SSS SAS ASA AAS Congruence Statement if necessary: Unit2‐6 page49 HL Not Enough Information 2. Prove which of the following triangles congruent if possible by filling in the missing blanks: a. Given ≅ and ∥ Statements 1. ≅ 2. ∥ 3. ∡ 5. ∆ c. Given ≅ and Point A is the and ≅∡ ≅ 4. ≅ b. Given midpoint of Reasons ≅∆ Statements Reasons 1. Given 2. Given 3. Definition of Midpoint 4. Reflexive property of congruence 5. By steps 1,3,4 and SSS ∥ Statements 1. Given 2. Given 3. 4. 5. ∆ M.Winking Reasons Unit2‐6 ≅∆ page50 Prove the Isosceles Triangle Theorem and the rest of the suggested proofs. d. Given ∆ is isosceles and point R is the midpoint of Statements 1. ∆ is isosceles 2. Reasons ≅ 3. R is the midpoint of e. Given point I is the midpoint of point I is the midpoint of and 4. ≅ 5. ≅ 6. ∆ ≅∆ 7. ∡ ≅∡ Statements 1. I is the midpoint of 2. Reasons Definition of Midpoint 3. I is the midpoint of 4. f. Given ∡ and ≅ and ∡ 5. ∡ ≅∡ 6. ∆ ≅∆ are right angles Statements 1. ∡ &∡ are right angles 2. ≅ 3. ≅ 4. ∆ M.Winking Unit2‐6 ≅∆ page51 Reasons Prove the suggested proofs by filling in the missing blanks. ∥ g. Given and ∥ Statements 1. Reasons ∥ 2. 3. ∥ 4. 5. 6. ∆ h. Given ∥ ≅ ,∡ ≅∡ ≅∆ and Statements 1. Reasons ≅ 2. 3. 4. and Substitution Property 5. M.Winking 6. ≅ 7. ∥ 8. ∡ ≅∡ 9. ∡ ≅∡ 10. ∆ ≅∆ Unit2‐6 page52 Prove the suggested proofs by filling in the missing blanks. i. Given ∡ ≅∡ ,∡ ≅∡ and Point A is the midpoint of , ii. Statements 1. ∡ Reasons ≅∡ 2. ∆ is an isosceles triangle 3. ≅ 4. A is the midpoint of 5. j. bisects∡ . Given that and are radii of the Also, same circle with center A. 6. ∡ ≅∡ 7. ∆ ≅∆ Statements 1. Reasons bisects ∡ 2. Definition of Angle Bisector 3. Radii of the same circle are congruent. 4. Reflexive property of congruence 5. ∆ M.Winking ≅ Unit2‐6 ≅∆ page53 Prove the suggested proofs by filling in the missing blanks. i. Given: ∡ ≅∡ ≅ forms an isosceles triangle with base ≅∆ Prove: ∆ Statements Reasons 1. ∡1 ≅ ∡9 ≅ 2. 3. ∡2 ≅ ∡4 4. ∆ is an isosceles triangle 5. Isosceles Triangle Theorem 6. ∡6 ≅ ∡7 7. ∡2 ≅ ∡7 8. Prove the suggested proofs by filling in the missing blanks. k. Given: The circle has a center at point C forms an isosceles triangle with base Prove: ∆ Statements 1. ∆ is an isosceles triangle w/ base ≅ 2. ≅∆ Reasons 3. The circle is centered at point C ≅ 4. Reflexive Property of Congruence 5. 6. ∆ M.Winking Unit2‐6 ≅∆ page54